Multiscale Shape Research Articles

A multiscale stochastic finite element method on elliptic problems involving uncertainties

In this study a multiscale stochastic finite element method (MsSFEM) is developed to resolve scale-coupling stochastic elliptic problems, based on formulations of a stochastic variational approach and scale-bridging multiscale shape functions. By employing polynomial chaos expansions and Lagrange polynomials, stochastic Galerkin method and stochastic collocation method are introduced and combined with deterministic multiscale methods, i.e. Variational Multiscale Method and Multiscale Finite Element Method. The resulting spectral and pseudo-spectral stochastic finite element methods are incorporated into the fine- and coarse-mesh computation of the MsSFEM. A benchmark multiscale model and the numerical experiment based on the formulated MsSFEM are provided for illustration. It is expected that the proposed MsSFEM can act as a paradigm for solving of general stochastic partial differential equations involving multiscale stochastic data.

Analysis of Factors Influencing the Shear Deformation of Granular Materials

Abstract This study utilizes recent advances in image analysis techniques to conduct detailed multiscale shape analysis of four granular materials. In addition, X-ray analysis was used to measure the chemical composition of these granular materials. The shape and chemical composition results were used to investigate the factors influencing the shear deformation of granular materials tested using the direct shear device. The analysis involved determining the different energy components influencing the shear deformation of granular materials. The results show that the granular material angularity and texture are two different distinct properties that can be quantified accurately using the methods proposed in this paper. Texture influences the material “true” friction,” while angularity mostly affects the dilative behavior of the materials. The results also show that the energy dissipated in particle rearrangement is an important factor that should be considered in the analysis of shear deformation of granular materials. This dissipated energy increases with an increase in material angularity and a decrease in confinement. A relationship between the various components of the angle of internal friction and stress confinement is proposed in this study.

A graph-based approach for multiscale shape analysis

This paper presents two shape descriptors, multiscale fractal dimension and contour saliences, using a graph-based approach— the image foresting transform. It introduces a robust approach to locate contour saliences from the relation between contour and skeleton. The contour salience descriptor consists of a vector, with salience location and value along the contour, and a matching algorithm. We compare both descriptors with fractal dimension, Fourier descriptors, moment invariants, Curvature Scale Space and Beam Angle Statistics regarding to their invariance to object characteristics that belong to a same class (compact-ability) and to their ability to separate objects of distinct classes (separability).

A multi-scale shape descriptor for shape similarity search of 3D models

A multi-scale shape descriptor for shape similarity search of 3D models

POLYGONAL CURVE EVOLUTIONS FOR PLANAR SHAPE MODELING AND ANALYSIS

The paper develops a numerical method to analyze and model planar shapes via polygonal curve evolutions. Consider a smooth curve each point of which moves in the normal direction with speed equal to a function of the curvature and curvature derivatives at the point. Chosen the speed function properly, the evolving curve converges to a desired shape. Smooth curve evolutions are approximated by evolutions of polygonal curves. Discrete analogs of the curvature and its derivatives are considered and analyzed. We apply our approach to simplify curves for purposes of multiscale shape analysis, to model geodesics on a plane equipped with a Riemannian metric, to design of nonlinear splines satisfying various boundary conditions, and to detect objects in grey-scale images.

Active shape focusing

This paper presents a framework for hierarchical shape description which enables quantitative and qualitative shape studies at multiple levels of image detail. It allows the capture of the global object shape at higher image scales, and to focus it down to finer details at decreasing levels of image scale. A multi-scale active contour model, whose energy function is regularized with respect to underlying geometric image structure in a natural scale setting, is developed for the purpose of implicit shape extraction or regularization with respect to scale. The resulting set of shapes is formulated and visualized as a multi-scale shape stack for the investigation of shape changes across scales. We demonstrate the functionality of this framework by applying it to a set of true fractal structures, and to 3D brain MRI. The framework is shown to be capable of recovering the fractal dimension of the fractal shapes directly from their embedding image context. The equivalent measure on the medical images and its potential for medical shape analysis is discussed.

Neural cell classification by wavelets and multiscale curvature.

A new approach to automatic classification of retinal ganglion cells using multiscale techniques including the continuous wavelet transform, curvature, and standard pattern recognition techniques is described. Each neural cell is represented by its outer contour, and the wavelet transform is calculated from the complex signal defined by the aforementioned contour, leading to the so-called W-representation (Antoine et al. 1996). The normalized multiscale wavelet energy (NMWE) is used to define a set of shape measures associated with the number of details of the shape for a broad range of spatial scales. Next, the more discriminating NMWE coefficients are chosen through a feature ordering technique and fed to statistical classifiers. In addition, the normalized multiscale bending energy (NMBE) is discussed as a means of neural shape description for classification purposes based on the multiscale curvature, i.e. the curvegram, of the neural contour. It is shown that both shape descriptors are suitable for shape classification, presenting similar classification performance. In fact, NMBE has a slightly better recognition rate than NMWE in our experiments. On the other hand, NMWE is less computationally expensive than NMBE, presenting also the potentially useful property of allowing the use of more suitable different analyzing wavelets, depending on the problem under consideration. Therefore, both measures are related and provide a good framework for the design of neural cell description and classification. The methods described in this work have been successfully applied to the classification of two classes of cat retinal ganglion cells, namely alpha and beta (henceforth referred as alpha-cells and beta-cells, respectively), and three statistical classifiers were considered: minimum-distance, k-nearest neighbours and maximum likelihood. The mean recognition rates are near 90%, which is superior to the other shape measures considered. It is argued here that the proposed technique can be adopted as a new general methodology for multiscale shape analysis and recognition, being applicable also to other problems in biological shape characterization in neuroscience and general biomedical image analysis. In the context of analysis of shape complexity, the multiscale energies are coherent with subjective judgements by humans.

Multiscale Contour Segmentation and Approximation: An Algorithm Based on the Geometry of Regular Inscribed Polygons

This paper describes an algorithm to segment and approximate a contour into circular arcs and straight line segments. This algorithm finds an adequate set of circular arcs and straight line segments to describe the contour shape. The resulting description respects a large number of shape representation criteria much better than any previous method. These criteria support efficient, general-purpose, unique, shape preserving, local, robust, stable, invariant, multiscale, coherent, hierarchical, and multipart shape representations. The adequacy of the algorithm is a direct consequence of the formal computational theory on which it is based. The resulting description may depend on sensor accuracy, sampling step size, minimum and maximum contour curvatures, and scale. The algorithm is thus parameterized using a small number of thresholds related to these data set and task properties. Predictions regarding the ground-truth performance of the algorithm can be made given some knowledge of the above properties. Experimental comparative results are presented for contour data of varying complexity.

Shape characterization with the wavelet transform

We present a new approach to the problem of two-dimensional multiscale shape representation and analysis, based on the one-dimensional continuous wavelet transform (CWT). The shape is represented by the complex signal that describes its boundary, and the CWT is applied to this signal, leading to the so-called W-representation. Wavelet theory provides the W-representation with several properties that are generally required from shape representation frameworks. In addition, we introduce algorithms for extracting meaningful information about the shape from its W-representation, for instance, detection of dominant points and shape partitioning, natural scales analysis, and fractal-based analysis. The algorithms that accomplish these tasks are tested on shapes obtained from synthetic and real images. Thus the W-representation yields a unified approach to a number of important problems of shape characterization for purposes of machine vision.

A Theory of Multiscale, Torsion-Based Shape Representation for Space Curves

It is argued that space curves are useful for representing 3-D surfaces and objects. This paper introduces a multiscale shape representation, referred to as the torsion scale space image(or TSS), for space curves. Experiments show that the representation is robust and suitable for recognition of noisy curves at any scale or orientation. The method rests on the concept of describing curves at varying levels of detail using invariant geometric features. Three different ways of computing the representation, each with different properties, are described here. A two-phase matching algorithm consisting of TSS matching followed by transformation parameter optimization demonstrates the usefulness of the representations in recognition tasks. The process of describing curves at increasing levels of abstraction is referred to as the evolutionof those curves. Several evolution properties of space curves are described in this paper. Together, these results provide a sound theoretical foundation for the representation methods introduced here.

Discrete morphological size distributions and densities: estimation techniques and applications

Morphological size distributions and densities are frequently used as descriptors of granularity or texture within an image. They have been successfully employed in a number of image processing and analysis tasks, including shape analysis, multiscale shape representation, texture classification, and noise filtering. In most cases however it is not possible to analytically compute these quantities. In this paper, we study the problem of estimating the (discrete) morphological size distribution and density of random images, by means of empirical as well as Monte Carlo estimators. Theoretical and experimental results demonstrate clear superiority of the Monte Carlo estimation approach. Examples illustrate the usefulness of the proposed estimators in traditional image processing and analysis problems.

Pattern spectrum and multiscale shape representation

The results of a study on multiscale shape description, smoothing and representation are reported. Multiscale nonlinear smoothing filters are first developed, using morphological opening and closings. G. Matheron (1975) used openings and closings to obtain probabilistic size distributions of Euclidean-space sets (continuous binary images). These distributions are used to develop a concept of pattern spectrum (a shape-size descriptor). A pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals, as well as for discrete images, by developing a discrete-size family of patterns. Large jumps in the pattern spectrum at a certain scale indicate the existence of major (protruding or intruding) substructures of the signal at the scale. An entropy-like shape-size complexity measure is also developed based on the pattern spectrum. For shape representation, a reduced morphological skeleton transform is introduced for discrete binary and graytone images. This transform is a sequence of skeleton components (sparse images) which represent the original shape at various scales. It is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components; in addition, two-way correspondences are established among the degree of shape smoothing via multiscale openings or closings, the pattern spectrum zero values, and the elimination or nonexistence of skeleton components at certain scales.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>